5 problems

Let $P_1,\dots,P_m\in\mathbf{Z}[y]$ be polynomials and let $0\leq i\leq m$. Consider the polynomial progression … An algebraic relation among these entries is an identity … where…

Let $P_1,\dots,P_s\in\mathbb{Z}[x]$ be linearly independent polynomials that vanish at zero. For $\omega\in\{0,1\}^s$, write $\omega=(\omega_1,\dots,\omega_s)$ and…

Let $P_1,\dots,P_s\in\mathbb{Z}[x]$ be linearly independent polynomials that vanish at zero. Let $f_\omega:\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}$ be one-bounded functions, and define…

Let $P_1,\dots,P_s\in\mathbb{Z}[x]$ be linearly independent polynomials that vanish at zero. Let $f_\omega:\mathbb{Z}/N\mathbb{Z}\to\mathbb{C}$ be one-bounded functions for each…

Let $t\in\mathbb{N}_+$ and let $\vec{P}\in\mathbb{R}[x,y]^{t+1}$ be an integral polynomial progression. For each $0\leqslant i\leqslant t$, let $\mathcal{HK}_i(\vec{P})$,…